Stochastic periodic orbits in fast-slow systems with self-induced stochastic resonance

Jinjie Zhu,Hiroya Nakao

doi:10.1103/physrevresearch.3.033070

Abstract

Noise is ubiquitous in various systems. In systems with multiple timescales, noise can induce various coherent behaviors. Self-induced stochastic resonance (SISR) is a typical noise-induced phenomenon identified in such systems, wherein noise acting on the fast subsystem causes stochastic resonancelike boundary crossings. In this paper, we analyze the stochastic periodic orbits caused by SISR in fast-slow systems. By introducing the notion of the mean first passage velocity toward the boundary, a distance matching condition is established, through which the critical transition position of boundary crossing can be calculated. The theoretical stochastic periodic orbit can be accordingly obtained via gluing the dynamics along the slow manifolds. It is shown that the theoretical predictions are in excellent agreement with the results of Monte Carlo simulations for a piecewise linear FitzHugh-Nagumo system even for large noise. Furthermore, the proposed method is extended to the original FitzHugh-Nagumo system and also found to exhibit consistent accuracy. These results provide insights into the mechanisms of coherent behaviors in fast-slow systems and will shed light on the coherent behaviors in more complex systems and large networks.

Highlights

Once we obtain the critical transition positions, the other parts of the stochastic periodic orbit can be determined by the deterministic slow dynamics along the left and right branches
The theoretical stochastic periodic orbits and the results of Monte Carlo simulations for various noise strengths are displayed in Fig. 7
The theoretical stochastic periodic orbit is obtained by gluing the motions along the slow manifolds

Summary

INTRODUCTION

Yamakou and Jost [26] further clarified the allowed interval for the potential difference and discussed the connection between inverse stochastic resonance and SISR These analyses required both the slow timescale and the noise strength to approach zero. To accurately obtain the stochastic periodic orbit in SISR oscillators, we define the mean first passage velocity and propose a condition that determines the critical transition position under the assumption that the transition process is continuous. This assumption differs from the previous studies, where the transition is considered as instantaneous and determined via the timescalematching condition [16,17].

PIECEWISE LINEAR FITZHUGH-NAGUMO SYSTEM

STOCHASTIC PERIODIC ORBIT OF PWL-FHN SYSTEM

STOCHASTIC PERIODIC ORBIT OF FITZHUGH-NAGUMO SYSTEM

CONCLUSIONS AND DISCUSSION

Theory Simulation